What Is Cos ⁡ 45 ∘ \cos 45^{\circ} Cos 4 5 ∘ ?A. 3 2 \frac{\sqrt{3}}{2} 2 3 ​ ​ B. 1 2 \frac{1}{2} 2 1 ​ C. 1 2 \frac{1}{\sqrt{2}} 2 ​ 1 ​ D. 1 E. 1 3 \frac{1}{\sqrt{3}} 3 ​ 1 ​ F. 3 \sqrt{3} 3 ​

Introduction

In mathematics, trigonometry is a branch that deals with the relationships between the sides and angles of triangles. One of the fundamental concepts in trigonometry is the cosine function, which is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. In this article, we will explore the value of $\cos 45^{\circ}$, which is a fundamental trigonometric ratio.

Understanding the Cosine Function

The cosine function is a periodic function that oscillates between -1 and 1. It is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. The cosine function is denoted by the symbol $\cos \theta$, where $\theta$ is the angle in question. In this case, we are interested in finding the value of $\cos 45^{\circ}$.

The Unit Circle

To find the value of $\cos 45^{\circ}$, we can use the unit circle, which is a circle with a radius of 1 centered at the origin of the coordinate plane. The unit circle is a fundamental concept in trigonometry, and it is used to define the values of the trigonometric functions.

Finding the Value of $\cos 45^{\circ}$

To find the value of $\cos 45^{\circ}$, we can use the unit circle. We start by drawing a line from the origin to the point on the unit circle that corresponds to an angle of $45^{\circ}$. This line represents the adjacent side of the right-angled triangle. The length of this line is equal to the cosine of the angle.

Using the Pythagorean Identity

We can use the Pythagorean identity to find the value of $\cos 45^{\circ}$. The Pythagorean identity states that $\sin^2 \theta + \cos^2 \theta = 1$, where $\theta$ is the angle in question. Since we are interested in finding the value of $\cos 45^{\circ}$, we can substitute $\theta = 45^{\circ}$ into the Pythagorean identity.

Solving for $\cos 45^{\circ}$

Using the Pythagorean identity, we can solve for $\cos 45^{\circ}$. We start by substituting $\theta = 45^{\circ}$ into the Pythagorean identity: $\sin^2 45^{\circ} + \cos^2 45^{\circ} = 1$. Since $\sin 45^{\circ} = \cos 45^{\circ}$, we can simplify the equation to $2 \cos^2 45^{\circ} = 1$.

Simplifying the Equation

We can simplify the equation by dividing both sides by 2: $\cos^2 45^{\circ} = \frac{1}{2}$. Taking the square root of both sides, we get $\cos 45^{\circ} = \pm \sqrt{\frac{1}{2}}$.

Evaluating the Square Root

The square root of $\frac{1}{2}$ is equal to $\frac{1}{\sqrt{2}}$. Therefore, we can simplify the expression for $\cos 45^{\circ}$ to $\cos 45^{\circ} = \pm \frac{1}{\sqrt{2}}$.

Choosing the Correct Sign

Since the cosine function is positive in the first quadrant, we can choose the positive sign for $\cos 45^{\circ}$. Therefore, the value of $\cos 45^{\circ}$ is $\cos 45^{\circ} = \frac{1}{\sqrt{2}}$.

Conclusion

In conclusion, the value of $\cos 45^{\circ}$ is $\frac{1}{\sqrt{2}}$. This is a fundamental trigonometric ratio that is used in a wide range of mathematical and scientific applications. Understanding the value of $\cos 45^{\circ}$ is essential for solving problems in trigonometry and other areas of mathematics.

Final Answer

The final answer is: $\boxed{\frac{1}{\sqrt{2}}}$

Introduction

In our previous article, we explored the value of $\cos 45^{\circ}$, which is a fundamental trigonometric ratio. In this article, we will answer some of the most frequently asked questions about $\cos 45^{\circ}$.

Q: What is the value of $\cos 45^{\circ}$?

A: The value of $\cos 45^{\circ}$ is $\frac{1}{\sqrt{2}}$.

Q: Why is $\cos 45^{\circ}$ equal to $\frac{1}{\sqrt{2}}$?

A: The value of $\cos 45^{\circ}$ is equal to $\frac{1}{\sqrt{2}}$ because of the properties of the unit circle and the Pythagorean identity.

Q: What is the unit circle?

A: The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. It is a fundamental concept in trigonometry and is used to define the values of the trigonometric functions.

Q: How is the unit circle used to find the value of $\cos 45^{\circ}$?

A: The unit circle is used to find the value of $\cos 45^{\circ}$ by drawing a line from the origin to the point on the unit circle that corresponds to an angle of $45^{\circ}$. The length of this line is equal to the cosine of the angle.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states that $\sin^2 \theta + \cos^2 \theta = 1$, where $\theta$ is the angle in question.

Q: How is the Pythagorean identity used to find the value of $\cos 45^{\circ}$?

A: The Pythagorean identity is used to find the value of $\cos 45^{\circ}$ by substituting $\theta = 45^{\circ}$ into the equation. This gives us $2 \cos^2 45^{\circ} = 1$, which can be simplified to $\cos^2 45^{\circ} = \frac{1}{2}$.

Q: What is the final answer for $\cos 45^{\circ}$?

A: The final answer for $\cos 45^{\circ}$ is $\boxed{\frac{1}{\sqrt{2}}}$.

Q: Why is $\cos 45^{\circ}$ an important trigonometric ratio?

A: $\cos 45^{\circ}$ is an important trigonometric ratio because it is used in a wide range of mathematical and scientific applications, including physics, engineering, and computer science.

Q: How can I use $\cos 45^{\circ}$ in real-world applications?

A: $\cos 45^{\circ}$ can be used in real-world applications such as calculating the height of a building, the distance between two points, and the angle of a triangle.

Conclusion

In conclusion, $\cos 45^{\circ}$ is a fundamental trigonometric ratio that is used in a wide range of mathematical and scientific applications. Understanding the value of $\cos 45^{\circ}$ is essential for solving problems in trigonometry and other areas of mathematics.

Final Answer

The final answer is: $\boxed{\frac{1}{\sqrt{2}}}$